Multilevel Filtering Elliptic Preconditioners

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Abstract
A class of preconditioners for elliptic problems built on ideas borrowed from the digital filtering theory and implemented on a multilevel grid structure is presented. These preconditioners are designed to be both rapidly convergent and highly parallelizable. The digital filtering viewpoint allows for the use of filter design techniques for constructing elliptic preconditioners and also provides an alternative framework for understanding several other recently proposed multilevel preconditioners. Numerical results are presented to assess the convergence behavior of the new methods and to compare them with other preconditioners of multilevel type, including the usual multigrid method as preconditioner, the hierarchical basis method, and a recent method proposed by Bramble–Pasciak–Xu.

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